Finite Group Databasebeta

← #120905

Subgroups — #120905

Order 512.

423 conjugacy classes of subgroups.

The subgroup lattice has 329 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21364C211yesmaximal, characteristic
2#8105C211yesmaximal, characteristic
3#8085C211yesmaximal, characteristic
4#8052C211yesmaximal, characteristic
5#7984C211yesmaximal, characteristic
6#7977C211yesmaximal, characteristic
7#7949C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#432C2211yescharacteristic
12#2285C2211yescharacteristic
13#2283C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1948—21no—
16#1947—21no—
17#1854—22no—
18#1776—22no—
19#1305—22no—
20#1302—22no—
21#1297—22no—
22#1289—22no—
23#1288C414yescomplemented
24#1285—21no—
25#1285—21no—
26C2×#117—21no—
27C2×#117—21no—
28C22×C42C2311yescharacteristic
29#432—21no—
30#432—21no—
31#432—21no—
32#432—21no—
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432—21no—
38#432—21no—
39#432C2×C412yes—
40#432Q812yes—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45C2×#107—22no—
46C2×#106—22no—
47#423—21no—
48#423—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8Q811yescharacteristic
51C2×C4⋊C8Q811yescharacteristic
52#407—21no—
53#407D412yes—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C4×M16—21no—
57C4×M16C2×C412yes—
58C2×#97—21no—
59C2×#97D412yes—
60#384—21no—
61#383C2×C412yes—
62#382—21no—
63#382—21no—
64#381—21no—
65#380—21no—
66#380—21no—
67#380D412yes—
68C4×C4⋊C4—21no—
69C4×C4⋊C4—21no—
70C4×C4⋊C4—21no—
71C4×C4⋊C4—21no—
72C4×C4⋊C4D411yescharacteristic
73C4×C4⋊C4D411yescharacteristic
74C4×C22⋊C4—21no—
75C4×C22⋊C4—21no—
76C4×C22⋊C4—21no—
77#376—21no—
78#376—21no—
79C43—21no—
80#340—22no—
81#340—22no—
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4#4111yescharacteristic
85C2×M16—22no—
86C2×M16—22no—
87C22×C8—22no—
88C22×C8—21no—
89C22×C8—21no—
90#117—42no—
91#117—42no—
92#117—82no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—41no—
105C2×C42—41no—
106C2×C42—41no—
107C2×C42—41no—
108C2×C42C22×C411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42C2×Q811yescharacteristic
112C2×C42C2×Q811yescharacteristic
113C2×C42#4111yescharacteristic
114C2×C42#4111yescharacteristic
115C2×C42#4111yescharacteristic
116C2×C42#4111yescharacteristic
117C2×C42#4111yescharacteristic
118C2×C42#4111yescharacteristic
119C2×C42#4111yescharacteristic
120#107—44no—
121#106—44no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—21no—
125C4⋊C8—21no—
126C4⋊C8—21no—
127C4⋊C8—21no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8—41no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140#97—22no—
141#97—22no—
142#97—41no—
143#97—41no—
144#97—41no—
145#97C4⋊C414yes—
146C4×C8—22no—
147C4×C8—41no—
148#95—22no—
149#95—22no—
150#95—22no—
151#95—21no—
152#95—21no—
153#95—22no—
154#95—41no—
155#95—41no—
156#95—41no—
157C24#12111yescharacteristic
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—81no—
167C22×C4#11711yescharacteristic
168C22×C4C4×D411yescharacteristic
169C22×C4C4×D411yescharacteristic
170C22×C4#12011yescharacteristic
171C22×C4#12111yescharacteristic
172C22×C4#12111yescharacteristic
173C22×C4C22⋊Q811yescharacteristic
174C22×C4C22⋊Q811yescharacteristic
175C22×C4#12311yescharacteristic
176C22×C4#12311yescharacteristic
177C22×C4#12311yescharacteristic
178C22×C4#12311yescharacteristic
179C22×C4#12411yescharacteristic
180C22×C4#12411yescharacteristic
181C22×C4#12511yescharacteristic
182C22×C4#12511yescharacteristic
183C22×C4#12611yescharacteristic
184C22×C4C4⋊Q811yescharacteristic
185M16—82no—
186M16—82no—
187C2×C8—24no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—42no—
194C2×C8—81no—
195C2×C8—81no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—81no—
203C4⋊C4—81no—
204C4⋊C4—82no—
205C4⋊C4—82no—
206C22⋊C4—44no—
207C22⋊C4—81no—
208C22⋊C4—81no—
209C22⋊C4—82no—
210C42—44no—
211C42—81no—
212C42—81no—
213C42—82no—
214C42—82no—
215C42—82no—
216C42C2×C4⋊C411yescharacteristic
217C42#11711yescharacteristic
218C42C4×D411yescharacteristic
219C42C4×Q811yescharacteristic
220C42C22⋊Q811yescharacteristic
221C42C22⋊Q811yescharacteristic
222C42C22⋊Q811yescharacteristic
223C42#12311yescharacteristic
224C42#12411yescharacteristic
225C42#12411yescharacteristic
226C42#12511yescharacteristic
227C42#12511yescharacteristic
228C42#12511yescharacteristic
229C42#12611yescharacteristic
230C42#12611yescharacteristic
231C42#12611yescharacteristic
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23#38811yescharacteristic
237C23#39411yescharacteristic
238C23#39811yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4#38211yescharacteristic
275C2×C4#38511yescharacteristic
276C2×C4#38711yescharacteristic
277C2×C4#38711yescharacteristic
278C2×C4#38811yescharacteristic
279C2×C4#38911yescharacteristic
280C2×C4#39311yescharacteristic
281C2×C4#39711yescharacteristic
282C2×C4#39811yescharacteristic
283C2×C4#40011yescharacteristic
284C2×C4#40011yescharacteristic
285C2×C4#40011yescharacteristic
286C8—44no—
287C8—82no—
288C8—82no—
289C8—81no—
290C8—81no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—81no—
300C22—81no—
301C22#144211yescharacteristic
302C22#204811yescharacteristic
303C22#204911yescharacteristic
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—82no—
316C4—82no—
317C4—162no—
318C4—162no—
319C4#203111yescharacteristic
320C4#203111yescharacteristic
321C2—21no—
322C2—21no—
323C2—41no—
324C2—41no—
325C2#1127411yescharacteristic
326C2#1127711yescharacteristic
327C2#1128211yescharacteristic
subgroupG/H
center, socleC22#1442
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
23. #1288C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#213642
#81052
#80852
#80522
#79842
#79772
#79492